Hodge Representations

Abstract

Hodge representations were introduced by Green-Griffiths-Kerr to classify the Hodge groups of polarized Hodge structures, and the corresponding Mumford-Tate subdomains of a period domain. The purpose of this article is to provide an exposition of how, given a fixed period domain $\mathcal{D}$, to enumerate the Hodge representations corresponding to Mumford-Tate subdomains $D\subset \mathcal{D}$. After reviewing the well-known classical cases that $\mathcal{D}$ is Hermitian symmetric (weight $n=1$, and weight $n=2$ with $p_g=h^{2,0}=1$), we illustrate this in the case that $\mathcal{D}$ is the period domain parameterizing polarized Hodge structures of (effective) weight two Hodge structures with first Hodge number $p_g=h^{2,0}=2$. We also classify the Hodge representations of Calabi-Yau type, and enumerate the horizontal representations of CY 3-fold type. (The "horizontal" representations those with the property that corresponding domain $D\subset \mathcal{D}$ satisfies the infinitesimal period relation, a.k.a. Griffiths' transversality, and is therefore Hermitian.)

References in corpus (2)

• Examples of Calabi-Yau threefolds parametrised by Shimura varieties
• Hodge Representations

Cited by in corpus (2)

• Hodge Representations
• Hodge Representations of Calabi-Yau 3-fold Type